As early as 1537 Pedro Nunes,2 the eminent Portuguese cosmographer and mathematician, proposed and solved instrumentally upon a globe the problem of finding the latitude by two altitudes of the sun and the angle made by the azimuth circles passing through the sun, when the altitudes were taken.
Later on, when “watches” became more accurate, Robert Hues3 in 1592 also solved upon a globe the problem of finding the latitude by two altitudes and the elapsed time and thereby established the general problem of navigation, but the longitude at sea could not be found with reasonable accuracy for the lack of a “chronometer” or a perfect time-keeper.
Of Facio’s Problem4 “In the year 1728 Nicholas Facio Duillier, F.R.S.5 published a pamphlet, entitled Navigation Improved, in which he considered, more at large than had been done before,
1 Published by Thomas Groom & Co. Boston, 1843. 6th edition in 1866.
2 Tralado da sphere com a Theorica do Sol e da Lua. Lisbon, December 1, 1537.
1 Tradatus de globis et eorum usu. London, 1592. Dutch translation in 1597.
4 Taken from The Elements of Navigation, by John Robertson, F.R.S., late librarian of the Royal Society. 6th Edition, London, 1796, vol. II, pp. 345-47. Of these “Elements”—the leading work on Navigation of the eighteenth century—there are seven editions dated 1754 (a perfect copy of which the writer
the problem for finding the latitude from two observations of the sun’s altitude out of the meridian, with the time between them, which he states in the following terms:”
Problem
To find the latitude, by two altitudes of the sun, and the time between them, supposing the observations to have been made from the same place, or from different places, whether at sea or land. And likewise to find the hours, and the sun’s azimuths, at the times of both observations.
Thus he proposes expressly to distinguish between the case, when the two observations are made in the same place, and when, by the motion of the ship, in the interval between them, they are made under different zeniths; which does not appear to have been attempted before. He also extends the interval of time between the observations farther than is usually done, not confining them within the compass of one day;
Fig. 1
Projected upon the plane of the Equator
but makes them so remote, that the change in the sun’s declination may be necessary to be taken notice of.
His method of proceeding is this:
Supposing P the elevated pole, A and B the two places of the sun, and AP, PB, AB, arcs of great circles passing through the points P, A, and B. Then if PA be the distance of the sun from the pole P in the first observation, and PB its distance in the second, the angle APB corresponds to the time between the observations, whether that angle represents the whole interval of time or only its difference from one or more entire days, Z also denoting the zenith of the place, when both the observations are under the same.
Here, if the interval of time be large, AP may be different from PB. Therefore in the triangle APB from AP, BP, and the angle APB, he finds the side AB\ then from the three sides AB, BP, and AB, either of the angles PAB, PBA may be known; also from AZ, ZB and AB in the triangle AZB, either of the angles ABZ, BAZ; and in the last place, from these angles with ABP, or BAP, may be deduced the angle ZAP or ZBP, whence in either of the triangles PAZ or PBZ, PZ may be computed, the distance of the zenith Z from the pole.
He likewise further proposes to compute the angles APZ, BPZ for assigning the distance of each observation from noon, in order to determine the time of the day when they were respectively made.
All these trigonometrical operations he performs by one axiom (See Art. 251 Book IV) in the doctrine of spherical triangles, which had been applied by others also, not only to the finding of an angle from three sides given, but likewise to the computing of the third side from two sides, and an angle between them.
found in the Franklin Institute in Philadelphia), 1764, 1772 (a copy of which it was the writer’s great pleasure to present to the library of the U. S. Naval Academy, Annapolis, Md.), 1780, 1784, 1796 and 1805 (not mentioned in the Encyclopedia Britannica, 11th ed.). From the 2nd edition on, these "Elements” contain a very fine history of navigation, up to 1750, by the learned Dr. James Wilson.
Jerome Lalande, the famous French astronomer, declared in his Abrtgt de Navigation, (Paris, 1793): “C’est le livre le plus complet et le plus utile dans la Marine Anglaise."
5 Nicholas Facio (J. Nicholas Facio de Duillier, in the Record of the Royal Society of London, 3rd ed. 1912), a Swiss mathematician of Italian descent and a prot6g6 of Newton, was born at Basel in 1664 and died at Worcester in 1753. He was a Fellow of the Royal Society since 1687. In 1704 he was granted a patent for the application of jewels to pivot holes of watches and clocks.
But when the ship is in motion, Z is not to be considered as the zenith of the ship in either observation. However, in the first observation the ship’s zenith will always be in a circle described on A, as a pole, and with the interval AZ,8 suppose in x, and in the other observation will be in a circle described to the pole B with the interval BZ, suppose in y, the arch xy denoting the ship’s course, making with the meridian of the first observation the angle Pxy.
This angle Pxy, and the length of the are xy are to be assigned by the compass and log-line.
Now when the ship has been in motion during the interval between the observations, and these two additional data are added to the altitudes of the sun, and the interval of time, Facio proceeds thus:
Having found the point Z, as before, he computes in the triangle AZP, from AP, AZ, PZ, the angle AZP; also in the triangle BZP from BP, BZ, PZ, the angle BZP, and from the angles AZP, BZP, the angle AZB, which (the angles AZx and BZy being both right) will be equal toy Z x.
Then supposing the distance run x y to be so small an are that the triangle x Z y may be considered as rectilinear without any sensible error; and also that the angle x P Z is so small that x y meeting P Z in d, the angle P v y may be assumed as not sensibly differing from the angle of the ship’s course; whence the angle BZP having been computed, and thence the angle y Z v known (BZy being a right angle) from the angles y Z v and Z v y the angle Z y x, which is the complement of the angle which the ship’s course makes with BZ, will be known; then from the angles x Z y, Z y x, together with the side y x, he finds the side Z x; and lastly, in the triangle P x Z from x Z, ZP, and the angle xZP, the side Px is found for the complement of the ship’s latitude at the first observation; as also the angle Z P x, and thence the angle A P x, the distance of the observation with the sun in A from the meridian in which that observation was made.
Fig. 2
Projected upon the plane of the Equator
And further, Facio considering this computation, though it came very near the truth, as an approximation only, from the assumption of the angles yxP, and Z v x, as equal and the triangle Z x y as rectilinear, he proceeds to correct the latitude, and the angle x P Z thus found, as follows:
He assumes at pleasure two values for Z x (or rather for the are of a great circle passing through Z and x) one somewhat greater, and the other somewhat less, than the value of Z x now computed. Then for each value of Z x, he finds in the triangle P x Z from the sides P Z, Z x, and the angle P Z x, the side P x and angle x P Z and > in the triangle Pxy from P x now found with x y, and the angle Pxy given from the course and distance run by the ship from x to y, he computes P y, and the angle x P y. Then from the angles x P Z, x P y finding the
6 This is the fundamental principle of modem sea and air navigation. In my “Tables” I expressed it as follows: “When a navigator at a given instant of Greenwich (known by a chronometer regulated to mean or sidereal time) observes the altitude of a celestial body, he determines ipso facto on the celestial sphere a small circle passing through his zenith.” In Facio's days there were no “chronometers” in existence and the problem was essentially a “latitude” problem. Some authors, including the writer, gave Captain Sumner credit for its discovery, but as we see it was not considered a novelty one hundred years before Captain Sumner’s time.
angle y P Z, and thence y P B, at length in the triangle B P y from P B, P y, and the angle B P y computes B y.
If B y, thus computed, comes out equal to B Z, then is the value of Z x, whence it is deduced, rightly assumed; but if the value in both the computations differ from the truth, P x and the angle Z P x are to be corrected by the simple rule of false position.7
But this method of Facio is attended with an arduous calculation, even though the second correction, which will always be but small, should be neglected. And the expedient proposed by Mr. Graham (Philosophical Transactions, No. 435) of observing by an azimuth compass, the angle which the ship’s course makes with the azimuth circle of the sun in one of the observations, which has been explained above (Book IX. Article 76) is much more simple.”
On December 9, 1731, Mr. Richard Graham, F.R.S., laid before the Royal Society: “The Description and Use of an Instrument for taking the Latitude of a place at any time of the Day” and in the Phil. Trans. No. 435 (December, 1734), page 450 he says: “Mr. Fatio [sic,] F.R.S. (in the year of 1728) proposed a method for finding the latitude, from two or more observations of the sun (or stars) at any time, the distance of the said observations in time, being given by a watch; but as his method requires a vast number of computations, and a great deal of skill in spherical trigonometry, it has very seldom been made use of, and never but by good mathematicians.”
Graham’s “instrument” was simply a globe properly fitted for solving the above problem and others such as the time of day at sea, etc., and, as Robertson says, “may indeed give the latitude within some minutes of the truth with ease and expedition” (loc. cit. vol. II, page 287). Like Facio, in the figures accompanying his exposition, he drew the “arches” of position on his globe to find the observer’s zenith.
It was not until the year of 1740 that Cornelius Douwes made known in Holland his approximate solution of this problem by utilizing the latitude by account or dead-reckoning, and his then recently computed logarithmic solar tables found their way to England, in manuscript, through the courtesy of Richard Harrison, who published them in London in 1759.
These logarithmic solar tables were extensively used until the advent of the perfect time-keeper or “chronometer” by John Harrison, in the year of 1765, after more than thirty years of patient and laborious experiments.
Methods for finding both latitude and time (longitude) were afterwards propounded by such eminent scientists as Borda, Bowditch, Brunnow, Caillet pere, Delambre, Gauss, Lalande, etc.
Although Captain Sumner’s book does not contain any quotations (Bowditch® excepted), one can readily see that he was familiar with all the important works
7 Here we find very ably expressed the whole theory of the “method of calculated zenith distances or calculated altitudes” and also the use of “assumed positions.”
For an interesting study of the general theory of false position applied to the determination of the observer’s position, see Ledieu, Les Nouvelles Methodes de Navigation, Dunod, Editeur, Paris, 1877, pp. 101-07.
8 Under the title of “Nathaniel Bowditch,” Lieutenant G. W. Logan, U. S. Navy, gave us in these Proceedings for December, 1903, a concise biography of the great American ship-master and mathematician, whose "New American Practical Navigator” is still well known all over the world.
Lieutenant Logan quotes at length from the preface to the original edition, issued in 1802, in which Bowditch tells the story of its origin. He shows how much he owed to John Robertson’s “Elements of Navigation”, to Dr. Maskelyne’s “Requisite Tables” and to John Hamilton Moore’s “New Practical Navigator”, the first American Edition of which was prepared under Bowditch’s own direction and published in 1799.
on navigation, due to the masterly way in which he later handled, in his book, his discovery of “the great value of a single altitude of the sun, when the latitude is uncertain” (6th edition, pp. 77-78) and explained “the principles upon which this method depends” (pages 52 et seq.).
The above expose emphasizes the great service rendered to all navigators by Captain Sumner’s happy discovery and a reference to his memory should be found in all works of modern sea and air navigation.
We will not attempt to go through Captain Sumner’s work, which should be carefully read by all students of navigation, but we will follow an interesting article by Lord Kelvin (then Sir William Thomson) in the Proceedings of the Royal Society, vol. XIX, London, 1871, p. 259, entitled: “On the Determination of a Ship’s Place from Observations of Altitude.” (Received February 6, 1871).8
Our purpose now is to show that the “newest” sea and air navigation by inspection had its birth four years before the so-called “new” navigation made its appearance on board the French school-ship Renommie and that it is based upon simple principles over a 1,000 years old.9 A full description of its conception follows:
“The ingenious and excellent idea of calculating the longitude from two different assumed latitudes with one altitude, marking on a chart the points thus found, drawing a line through them, and concluding that the ship was somewhere on that line at the time of the observation, is due to Captain T. H. Sumner. It is now well known to practical navigators. It is described in good books on navigation, as, for instance, Raper’s (§§ 1009-14)…I learned it first in 1858 from Captain Moriarty, Royal Navy, on board H.M.S. Agamemnon.” He further says: “A little experience at sea suggests that it would be very desirable .... to abolish calculation, as far as possible, in the ordinary day’s work.”
Speaking of Commander Thomas Lynn’s Horary Tables for finding the Time by Inspection, London, 1827, he says: “Tables of this kind have been actually calculated and published but have not come into general use.
“It has occurred to me, however, that by dividing the problem into the solution of two right-angled triangles, it may be practically worked out, so as to give the ship’s place as accurately as it can be deduced from the observations, without any calculation at all, by aid of a table of the solution of the 8,100 right-angled spherical triangles of which the legs are integral numbers of degrees.
“On 1870, May 16, afternoon, at 5h 42m Greenwich apparent time, the sun’s altitude was observed to be 32° 4'; to find the ship’s place, the assumed latitude being 54° North.
“The Nautical Almanac gives at 1870, May 16, 5h 42m Greenwich apparent time, the sun’s apparent declination N 19° 10'. [1]
from which we have the following:
Greenwich apparent time (in arc) 85° 30' ............................ 85° 30'
sun’s hour angle (1) 61 23 (2) 61 5
Diff.—Longitude (auxiliary) 24° 7'W............................ 24° 25'
sun’s altitude (observed 32° 4' .............................. 32° 4'
sun’s altitude (auxiliary) (1) 32 8 (2) 31 55
Diff. —4*' +9'
sun’s declination from N. A. 19° 10' ............................ 19° 10'
sun’s declination (auxiliary) (1) 19 11 (2) 18 42
Diff. -1' +28'
He works out two more examples with the same data, using assumed latitudes 10° North and 20° South and the altitudes respectively 30° 30' and 18° 35', drawing on a chart in each case the line of position from the assumed position by means of the difference between the auxiliary and the observed (true) altitudes and the auxiliary azimuths.
It seems that Lord Kelvin, having carefully perused Captain Sumner’s book, conceived the idea of computing the hour angle and the azimuth of a heavenly body (and incidentally its altitude) by means of non-logarithmic tables, and comparing this calculated altitude with the true altitude, by shifting the auxiliary Sumner straight line of position parallel to itself, he obtained the Sumner straight line of position for the true altitude.11 This idea was subsequently lost or forgotten, in view of the complications required to use his set of tables.
It was not until May, 1876—just a little over fifty years ago—that Lord Kelvin was able to publish his Tables for Facilitating Sumner’s Method at Sea.
Why his methods and tables did not succeed (although a second edition was published in London in 1886) has been very cleverly explained by Dr. Giuseppe Pesci, in the Rivista Marittima for January, 1909, p. 43. Many changes in the methods and and in the tables he devised were necessary before we could arrive at what the writer calls today: “the ‘newest’ sea and air navigation by means of inspection tables.”
- Lord Kelvin’s b is my c. He did not name the other leg or side, we call C.
- It is most interesting and gratifying to notice that Lord Kelvin’s idea, which the writer rendered practicable in 1908, has also been recently adopted in the new Hydrographic Office publications, known as H.O. 203 and H.O. 204 (dated 1923 and 1925), prepared by G. W. Littlehales, Hydrographic Engineer, but instead of reducing the observations by means of right-angled spherical triangle tables, he revived the excellent and useful Horary Tables of Lalande and Lynn (loc. cit.) and Lynn’s Azimuth Tables, London, 1829, adding many new arguments.
Lord Kelvin’s "auxiliary Sumner straight line of position” is called "preliminary Sumner line.” An integral degree of Latitude is used, to avoid interpolation, as was recommended in 1837 by Captain Sumner, by Lord Kelvin in 1870 and by many others later. Naturally no further reference is made to Marcq Saint Hilaire.
Admiral Marcq Saint Hilaire’s work dated, A bord de la Renommie, February, 1875, and published in the Revue Maritime et Coloniale,12 Vol. XLVI, August, 1875, pp. 341 and 714, did not appear until four years after Lord Kelvin presented his article to the Royal Society.
Photographic reduction of the page of Tables accompanying Lord Kelvin’s article, as presented to the Royal Society on February 6, 1871. The same type and disposition was used in his full set of Tables in 9 pages made public on November 11, 1875 and published in May, 1876.
u “Calcul du point observed; Methode des hauteurs estimties." Marcq Saint Hilaire used the estimated or dead-reckoning position and in his calculations he dropped a perpendicular from the body upon the meridian and worked the right-angled spherical triangle formulas by means of logarithms. He was the first to give us the “point Marcq” or “point approach”—the most probable position of the observer, when only one observation is available.
Did Marcq Saint Hilaire, in 1874, know about Lord Kelvin’s communication to the Royal Society made public on February 6, 1871?
Besides his reference and praise of Captain Sumner’s methods there are no references in Lord Kelvin’s works to such methods and tables as Towson’s,15 Deichmann’s,[2][3] [4] [5] Captain Bergen’s15 or Major-General Shortrede’s.16
Although all these tables were already in existence when Lord Kelvin presented his communication to the Royal Society, they all had in view principally the solution of problems in great circle sailing. Towson showed in 1861 how his tables could be used for finding azimuths. Captain Bergen having familiarized himself with Towson’s ex-meridian and great circle sailing tables showed that there were also many problems in navigation and nautical astronomy where minute accuracy was not required, and solved them by means of a set of very interesting spherical tables and a diagram.
Major General Shortrede after a long and tedious study of the right-angled spherical triangle tables or, as he calls those he published, “a Pantaspheric Table”, stated “it was not altogether the sort of table for common use at sea by anyone, and thereupon set about the construction of the azimuth tables now published”, following the deas already set forth in 1665 by Andrew Wakely (loc. cit.).
On p. 170 he states: “By means of a perpendicular, all cases of oblique spherics are reducible to those of right angles, and this table (or rather that of which this is an abridgment) has been used in the regular way for the common and troublesome case of “the sides and contained angle”, with results sufficiently exact, but obtained with so much trouble as evidently to be practically useless at sea.”
These quotations show the difficulties ahead when Lord Kelvin started.
A Russian edition of his “tables” was published by astronomer Kortazzi in Kronstadt in 1880, of which Collet issued a French edition, in Paris, in 1891. None of these were successful, however, because they followed too closely the original processes of Lord Kelvin, with additional complications.
In 1903 the writer published officially in Rio de Janeiro, in Portuguese, a simplified edition with quite a few changes and improvements, entitled “A Navegafao sem Log- arithmos” (navigation without logarithms), Imprensa Nacional.
Finally, after presenting to the American public his ideas in these Proceedings for December, 1908, the writer published in London, in 1910, the first edition of his Altitude and Azimuth Tables. A second edition appeared in 1912 greatly improved; reimpressions of this edition were made during the World War, in 1917 and 1918. The third edition, commemorating the centennial of Brazil’s Independence, appeared in 1924 with many additions and improvements. This edition was also prepared for aerial navigation, in view of their successful use in the air by Captain J. P. Ault, Commander of the non-magnetic ship Carnegie,11 in 1918.
Aquino’s “Newest” Sea and Air Navigation Tables.
The writer’s ultimate goal for the last twenty-four years18has always been to arrange the right-angled spherical triangle tables in such a way that they would present greater advantages than the well-known three argument inspection tables, where you enter with latitude, declination, and hour angle to find the altitude or the azimuth or both or where you enter with the latitude, declination, and altitude to find the hour angle and the azimuth or the position angle.
Due to the particular arrangement of these tables and the number of arguments two sets always have to be used: one set for the latitude and declination of the same names and another for the latitude and declination of contrary names.
Therefore, they are bulky, heavy, extensive, and expensive, if all the necessary arguments are carried in one, two or more volumes.
We believe we have attained our goal after twenty-four years of close study and patient investigation of the subject.
In only ninety pages—one page for each value of a—the “perpendicular”—and employing the latest methods available with very simple precepts, or none whatever, the writer hopes to show they represent the ideal navigation tables for use on the sea and in the air. They may be useful in many other problems requiring the use of right- angled spherics and this is the first time the tables appear complete. With two arguments above or below, the other three unknown quantities in the right-angled spherical triangle are given to the nearest tenth of a minute of are (within 3" of arc).
One of the pages, a = 56°, with which the examples, as originally given by Lord Kelvin can be worked out, shows the latest disposition of what the writer believes to be “ideal and fool-proof” tables.
The tables were computed by means of Collet’s19 seven place logarithms, from the following equations:
| sin d = cos a sin b |
| sin a = cos d sin t |
(1) | sin h = cos a sin B | (4) | sin a = cos h sin Z |
| cot t = cot a cos b |
| cot b = cot d cos t |
(2) | cot Z = cot a cos B | (5) | cot B = cot h cos Z |
| cot a = sin a tan b |
| cot a = sin d tan t |
(3) | cot ? = sin a tan B | (6) | cot ? = sin h tan Z |
- See ‘‘On Determination of Position of Airplanes by Astronomical Methods”. (Presented before the American Physical Society at Washington, April 25, 1919) and also Navigation of Aircraft by Astronomical Methods." By J. P. Ault. With 1 plate and 3 figures. 1926. (Extracted from Publication No. 175, Vol. V of the Carnegie Institution of Washington, pp. 315-37). He records “the first known instance of an airplane pilot being informed of his position by astronomical methods,” on September 23, 1918 (page 332).
- See an article in the Revista Maritima Brazileira for October, 1902, entitled Taboas par achar alturas e azimuths…” In this article and in my A Navegacao sent Logarithmos due reference and credit is given to Dr. Borgen for his timely article in the Annalen der Hydrographie, no. VII, July, 1902, 336, in which he revived Lord Kelvin’s tables and showed how they could be used to find the estimated altitude and azimuth in Marcq Saint Hilaire’s method. This was the basis of my 1903 edition.
- Tables de logarithmes, suivies d’un recueil de Tables Nautiques. Editeur Firmin Didot et Cie., Paris. 1883.
The notations contained in these equations are easily understood by looking at
the figure below, where the triangle of position is projected upon the plane of observer’s horizon.
Fig. 3
Projected upon the plane of the observer’s horizon
The following precepts show the relations between various elements of the triangle of position:
The above precepts given in order to avoid algebraic signs and arcs greater than 90°, may be reduced to the following, in practice:
Lat. and Dec. same names and H. A. less than 90°: C = L~b
Other cases……………………C = L+b
in view of the fact that the quadrant in which the body is observed is always known
a priori.
Equations (1), (2) and (3) were used to find Dec., H. A. and a from a and b (Lpper arguments).
We can also find Alt., Az. and ? from a and B.
We can also find co. H. A., co. Dec. and b from a and a.
We can also find co. Az., co. Alt. and B from a and ?.
Equations (4), (5) and (6) were used to find a, b and a from Dec. and H. A. (Lower arguments.).
We can also find c, 90°—a and H. A. from Dec. and a.
The interchanging of the upper arguments b and a (and B and ?), practically doubles the number of arguments, without increasing the volume of the tables.
The lower arguments are not really needed in our methods, except to obtain the approximate values of the upper arguments to facilitate the entry in the tables and to allow the use of them in connection with the other methods referred to on page 33.
This all shows the elasticity of the tables. As they are general and accurate in their nature they can be used in a great variety of ways.
The writer, naturally, has adapted them first for determining lines of position at sea and in the air and for the identification of celestial bodies, and second for great circle sailing and for the determination of lunar distances.
For these reasons the values appearing in our tables are the complements of those contained in Towson’s Great Circle Sailing Tables, as the problem of great circle sailing is secondary to the problem of determining lines of position at sea and in the air. Towson only gave the values to the nearest minute of arc, whereas we give them to the nearest tenth of a minute of are (within 3" of its true value).
The tables in their present form as shown here for the first time, slightly different from those appearing in these Proceedings for September, 1913, p. 1031, are available for three different practical methods for determining lines of position at sea and in the air:
- When use is made of the variation factors (Var. for 1') whereby the altitude difference can also be reduced to zero, in a great majority of the cases, allowing the line of position to be drawn right through the assumed position.
- When an integral degree of latitude is assumed and the position angle becomes necessary to correct the tabular altitude for the odd minutes of declination.
- When the use of' an integral degree of latitude is not desirable or practicable! which generally happens when the observed body is near the meridian or near the zenith—or when a change of latitude will give a small altitude difference or even reduce it to zero.
These tables are also available for use with any of the methods devised by Souil- lagouet,20 Delafon,21 Fuss,22 or Bertin.23
Examples
Lord Kelvin worked from data taken in the same a column or in adjacent a columns, according to the values of the data given. Only one value of a is necessary and all the data required can be found on one single page, as in the three argument tables.
First Method, a and C as integral numbers. Same example as Lord Kelvin’s on page 22, using the variation factors (Var. for 1’).
!0 Tables du Point Auxiliaire. Paris, July, 1891. Second and last edition at Toulouse, 1900. Table III was used by Lieutenant Leblanc in his non-logarithmic method.
21 Mithode rapide pour determiner les Droites et Courbes de Hauteur et faire le Point. Berger-Levrault et Cie., Editeurs, Paris, 1893.
“ Tablitzi dlya Nakashdeniya Visott i Azimutoff. Typography of the Imperial Academy of Sciences. Saint Petersburg, 1901.
” Extrait de la Table de Point SphCrique. Pour calculer 4 la Mer vite et sans erreur. Imprimerie Oberthur. Rennes, 1918.
This is the writer’s original modification of the process devised by Lord Kelvin in 1870, and, in a way, it is the simplest of all and was first described in these Proceedings for December, 1908.
A slight modification of the above method can be made, if we wish to reduce the Alt. Diff. to zero, by means of the variation factors (Var. for 1').
It can be most advantageously used when the azimuth is less than 60°, and the writer and many other navigators have used it, with good results, on azimuths up to 79°.
By adding or subtracting b and C from one another we find La. No precepts are necessary.
The “altitude-difference” is zero and the line of position can be drawn right through the assumed position: La. = 54°15:8N and Ga- = 24° 8:0W, as shown on our “Protractor Diagram”.
Second Method, a, b and C as integral numbers. Same example as above using an integral degree for the latitude. This is the writer’s modification of the First Method published in 1912, in the second edition of his Altitude and Azimuth Tables.
Third Method. Same example as above, using as arguments a and a, as integral numbers or a and (3, as integral numbers. In order to avoid interpolation C is always used as an integral number in the first process.
First Process, a and a as integral numbers. In this calculation we run down the co. Dec. column in order to find a tabular value nearest to the true co. Dec., which is 70° 50' and we find 70° 50:6. We take 6 = 35° 56:0 right alongside, to the left co. H. A. = 28° 38:5 or H.A. = 61° 21:5, and also outside a = 59°. Cis found as usual from L and b and the rest is the same as in all the methods.
Second Process, a and 0 as integral numbers. Instead of taking L = 53° 56:0, which combined with 6 = 35° 56:0 gave us C=18°, we can enter the tables inside, as
AQUINO'S PROTRACTOR DIAGRAM
When the “altitude-difference” is zero the line of position OB is drawn right through the assumed position: latitude 54° 15’.8 North and longitude 24° 8’.0 West. This permits the line of position to be drawn accurately and avoids the necessity of drawing a perpendicular. The dotted lines showing the direction of the sun from each assumed position need not be drawn.
Our “Protractor Diagram” is a position finding chart good for all latitudes and when it is used together with our “Graduated Triangle" the longitude scales are obtained immediately by simply adjusting the “graduated triangle” on the “Protractor Diagram.” See “Wrinkles in Plane Chart Methods” in these Proceedings, Vol. 40, No. 2, Whole No. 150; March-April, 1914.
before, and find 5 = 71° 28:9, co. Az. = 12° 5 :5 or 77° 54:5 and co. Alt. =57° 58:7 or Alt. = 32° 1:3 and also outside ? = 22°.
Combining 5 = 71° 28:9 (C=18° 31:1) with b = 35° 56:0, we would have La = 54° 27:1. By adding or subtracting b and C from one another we find La- No precepts are necessary in this process.
Above we have given three practical methods for finding lines of position at sea and in the air. The first two have had a wide acceptance all over the world, ever since the writer published them in 1910 and 1912.
By a happy coincidence in the first process of the third method above, the tabular co. Dec. is practically the same as the true co. Dec. and, in ordinary practice, especially in the air, no correction would have to be applied to the tabular altitude, and therefore M would not have to be found from a and ?.
The advantages of the new third method, with its two processes, do not become very apparent until the declinations and the altitudes are high (over 70°), as then to small changes of b or B in the tables correspond still smaller changes in co. Dec. or co. Alt. and the use of tedious interpolations is avoided and accuracy in the result is assured.
Page a = 5° O' is given to show this great advantage. Let the kind reader work out one of his high-altitude ex-meridians by means of this page and see the difference!
As the only calculation affecting the accuracy of the results (by which we mean the assumed altitude within 0:1, and the assumed longitude within 0:05 and the assumed Latitude within 0:1, whenever it is not an integral degree—exact in itself) is the correction of the tabidar altitude for the odd minutes of declination by means of the position angle M, it is always desirable to find a value of Dec. or co. Dec. nearest to the true value.
Precepts Linking L, b and C or L, c and B.
The use of our simple precepts in order to avoid the use of algebraic signs and arcs greater than 90° has always been encouraged by several distinguished mathematicians, and the writer succeeded in reducing the necessary precepts to only two covering the four possible cases—the same as those used for over a century in connection with the versine and haversine formulas.
Those who favor the adoption of 3 argument Tables point against these two very simple precepts.
The writer has succeeded in overcoming this by using TWO copies of his Tables— one for Lat. and Dec. Same names: H. A. less than 90° and one for contrary names and H. A. greater than 90°.
One copy would be marked “Lat. and Dec. Same Names: H. A. less than 90°” and B would be always L+c or C would be always L~b.
Another duplicate copy would be marked “Contrary Names and H. A. greater than 90°” and C would be always L+b or B would be always L~c.
In this simple way, with only ninety pages of tables we would obtain greater advantages than the three argument tables can offer. But is it really worth while to use two sets of tables to avoid our simple precepts?
Identification of Celestial Bodies
The problem of identifying celestial bodies is the reverse of the problem of determining altitude and azimuth. That is given: Alt., Az. and Lat. find Dec. and H. A.
Great Circle Sailing The problem of finding distance and course in great circle sailing may also be easily solved by these tables, because it is the same as determining altitude and azimuth. The distance corresponds to the zenith distance or co. Alt. and the course to the azimuth. The only difference is that the distance between the two given points can be greater than 90°, whereas the zenith distance cannot be greater than 90°.
If we wish to find a number of points along a great circle and the courses therefrom, it is better to determine the “latitude of its vertex.”
If we take as the “latitude of the vertex”, the small blackfaced number at the top (near a—the perpendicular), we will find the Latitude in column Dec., the Course in the column H.A., the longitude from vertex in column a and the distance in column c.
In this way we may use, to great advantage, the methods contained in Towson’s and Bergen’s Tables.
Lunar Distances
The problem of finding lunar distances (whenever necessary) is the same as the problem of determining zenith distance for lines of position or of finding distance in great circle sailing, as explained already, and the tables will give the lunar distance with the accuracy which can be obtained from seven decimal place logarithm.
All Other Problems Solved All the other problems in nautical astronomy depending upon the solution of right- angled spherical triangles can be easily solved by these tables by inspection, without interpolation.
Conclusions
When an author gives himself the real pleasure of reading old books on navigation, he finds that anything new, on this ever interesting subject, was always received and accepted by a few of the elite, until time had given its approval or it was forgotten.
Many, many years passed before the general acceptance of the Plane Traverse Tables, and even of the Nautical Almanac, as it is today.
Even forty-five years after the admirable invention of logarithms by Lord Napier in 1614, a distinguished mathematician, John Collins, published in London, in 1659, a wonderful little treatise on Navigation by the Mariners Plain Scale New Plain’d or, “A Treatise of Geometrical and Arithmetical Navigation”, in which he showed how all problems could be solved by his plain scale, without using logarithms or Gunter’s scales “both long and sliding.”
All throughout the fifteenth, sixteenth, seventeenth, and eighteenth centuries there was a wide variety of processes for solving all problems of navigation, but inspection tables such as Wakely’s, Lalande’s, etc., were making their way together with the Plane Traverse Tables and the Nautical Almanac, towards the final adoption of inspection methods.
And now we hope a greater interest will develop among navigators leading them to use “Inspection Tables,” of which we believe ours to be the safest, the “simplest and readiest” in solution and also the “most exact and the least expensive.”
As Lord Kelvin said twenty-four years ago: Navigators “ought to be spared the waste of time in making calculations, which can be ‘better done once for all’ by a single computer on dry land,” and we hope that by using these tables they will be spared "the waste of time” in these marvelous days of Millikan Rays, high-frequency radio and "radio vision.”
Appendix
"I have always thought that the chief motives which ought to induce a person to appear as a writer should be, either that he has something new to publish, or that he has arranged the parts of a known subject, in a method more regular and useful than had been done before, in either of these cases he cannot be a proper judge, unless he has seen the best pieces extant on that subject. On these principles I was led to examine what had been done by the different writers on navigation and having perused most of their books, of which I could get information, I had an opportunity of discovering the steps by which this art has risen to its present perfection, and consequently of knowing the most material parts of the history of its progress”.
The above words taken from the "Preface” of the third Edition of John Robertson’s Elements of Navigation, dated Nov. 1, 1772, express the present writer’s opinion and justify the program that was followed in the making of this article. The writer believes he has seen “at least, those of the most eminent authors already published”.
To those, however, who wish to follow this subject more closely, the writer recommends the careful study of the works of the two leading authorities on nautical astronomy in Great Britain and in the United States: H. B. Goodwin, M.A., F.R.A.S., Formerly Examiner in Nautical Astronomy at the Royal Naval College, Greenwich, and to H. M. Board of Education and Mr. G. W. Littlehales, Hydrographic Engineer, United States Hydrographic Office.
Special attention is called to Mr. Goodwin’s articles on “Great Circle Sailing” and “Spherical Traverse Tables and their Uses” in the Nautical Magazine, Vols. 68 and 69, London, 1900 and to Mr. Littlehales, The Development of Great Circle Sailing, U. S. Hydrographic Office. Second Edition, Washington, 1899, and his Patent No. 1,557,854 dated Oct. 20, 1925: "Method and means for finding geographical position in navigation”. The writer wishes to express his grateful thanks to the librarians of the New York Public Library, New York City, of the Library of Congress, and of the Smithsonian Institution and of the Navy Department, Washington, D. C., and of the Franklin Institute, Philadelphia, for their many courtesies while he was making his investigations, without which this article would be very incomplete.
8 See also "Navigation”: A lecture delivered in the City Hall, Glasgow, on Thursday, November 11, 1875; by Sir William Thomson, LL. D., D.C.L., F.R.S., etc. London and Glasgow, 1876 and Popular Lectures and Addresses by Sir William Thomson, in 3 vols. Macmillan and Co., London and New York, 1891. On p. 51 of Vol. Ill—Navigational Affairs—he describes “Sumner’s Method of Interpreting an Observation of Altitude,” very ably expressed.
9 The writer has been unable to ascertain if these methods were already used 4,200 years ago by "Noah, that Totius Orbis Thalassiarchus, or High Admiral of the Whole World, with his Ship, called, The Ark of Gopher, laden with a Cargo of the Whole Universe, after nearly six months voyage, arrived safely at Ararat, his Port of Discharge in Armenia.” (Joshua Kelly, The Modern Navigators’ Compleat Tutor. London, 1724, pp. i and ii.)
[2] John Thomas Towson, F.R.G.S., Tables to Facilitate the Practice of Great Circle Sailing. London,
1847—the pioneer of all tables of right-angled spherics. The writer has only the 6th ed. of 1861. Reprinted
in 1916. Sold by J. D. Potter, 145 Minories, London.
[5] A. H. Deichmann, New Tables to facilitate the Practice of Great Circle Sailing. London, 1857. Preface dated Hanover, March, 1856.
14 William Culley Bergen, Master in the Mercantile Marine. Spherical Tables and Diagram with Their Application to Great Circle Sailing and Various Problems in Nautical Astronomy. London, 1857. Preface dated Blyth, March 19, 1857.
14 Major General Robert Shortrede, F.R.A.S., a “Pantaspheric Table” in his Azimuth and Hour Angle for Latitude and Declination .... Together with a Great Circle Sailing Table. London, 1869. Preface dated February, 1868.